| L(s) = 1 | + 3·3-s − 6.30·5-s + 7·7-s + 9·9-s + 48.9·11-s − 2.60·13-s − 18.9·15-s + 136.·17-s + 45.2·19-s + 21·21-s − 38.1·23-s − 85.2·25-s + 27·27-s + 52.7·29-s − 14.7·31-s + 146.·33-s − 44.1·35-s + 333.·37-s − 7.82·39-s + 227.·41-s − 398.·43-s − 56.7·45-s − 184.·47-s + 49·49-s + 410.·51-s + 359.·53-s − 308.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.563·5-s + 0.377·7-s + 0.333·9-s + 1.34·11-s − 0.0556·13-s − 0.325·15-s + 1.95·17-s + 0.545·19-s + 0.218·21-s − 0.345·23-s − 0.682·25-s + 0.192·27-s + 0.337·29-s − 0.0856·31-s + 0.774·33-s − 0.213·35-s + 1.48·37-s − 0.0321·39-s + 0.865·41-s − 1.41·43-s − 0.187·45-s − 0.572·47-s + 0.142·49-s + 1.12·51-s + 0.932·53-s − 0.755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.141246333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141246333\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 6.30T + 125T^{2} \) |
| 11 | \( 1 - 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.60T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 99.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 836.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 201.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.06888665870260186297538466713, −11.65708535942331082089134661137, −10.16300705707623329432265438692, −9.275007092314261546216238069534, −8.109065288083680973331230213696, −7.36519219012300296816301509798, −5.90615384353472700633124666286, −4.32798627258125771807911710005, −3.25237628222443020654370158887, −1.31445216384837415690922545194,
1.31445216384837415690922545194, 3.25237628222443020654370158887, 4.32798627258125771807911710005, 5.90615384353472700633124666286, 7.36519219012300296816301509798, 8.109065288083680973331230213696, 9.275007092314261546216238069534, 10.16300705707623329432265438692, 11.65708535942331082089134661137, 12.06888665870260186297538466713
